1. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS

2. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS Hamiltonian formulation:

3. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS Hamiltonian formulation:

4. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS Hamiltonian formulation: Example: PII

5. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS Hamiltonian formulation: Isomonodromic deformations method (IMD): Example: PII

6. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS Hamiltonian formulation: Isomonodromic deformations method (IMD): Example: PII

7. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS Hamiltonian formulation: Isomonodromic deformations method (IMD): Example: PII

8. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS Hamiltonian formulation: Isomonodromic deformations method (IMD): Example: PII

9. In this course we shall see how to deduce the Hamiltonian formulation from the IMD.

10. Motivation: find the Hamiltonian structure of more complicated generalizations of the Painleve’ equations

11. In this course we shall see how to deduce the Hamiltonian formulation from the IMD. Motivation: find the Hamiltonian structure of more complicated generalizations of the Painleve’ equations Example: PII (2) What are p, p, q, q in this case? What is H? 1122

12. In this course we shall see how to deduce the Hamiltonian formulation from the IMD. Motivation: find the Hamiltonian structure of more complicated generalizations of the Painleve’ equations Example: PII (2) What are p, p, q, q in this case? What is H? 1122 Literature: Adler-Kostant-Symes, Adams-Harnad-Hurtubise, Gehktman, Hitchin, Krichever, Novikov-Veselov, Scott, Sklyanin…………….. Recent books: Adler-van Moerbeke-Vanhaeke Babelon-Bernard-Talon

13. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics)

14. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) M = phase space

15. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) M = phase space F (M) = algebra of differentiable functions

16. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) M = phase space F (M) = algebra of differentiable functions Poisson bracket: {, }: F (M) x F (M) -> F (M) {f,g} = -{g,f} skewsymmetry {f, a g+ b h} = a {f,g} + b {f,h} linearity {f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0 Jacobi {f, g h} = {f, g} h + {f, h} g Libenitz

17. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) M = phase space F (M) = algebra of differentiable functions Poisson bracket: {, }: F (M) x F (M) -> F (M) {f,g} = -{g,f} skewsymmetry {f, a g+ b h} = a {f,g} + b {f,h} linearity {f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0 Jacobi {f, g h} = {f, g} h + {f, h} g Libenitz Vector field X H associated to H e F (M): X H (f):= {H,f}

18. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) M = phase space F (M) = algebra of differentiable functions Poisson bracket: {, }: F (M) x F (M) -> F (M) {f,g} = -{g,f} skewsymmetry {f, a g+ b h} = a {f,g} + b {f,h} linearity {f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0 Jacobi {f, g h} = {f, g} h + {f, h} g Libenitz Vector field X H associated to H e F (M): X H (f):= {H,f} A Posson manifold is a differentiable manifold M with a Poisson bracket {, }

19. Recap on Lie groups and Lie algebras

20. Lie group G: analytic manifold with a compatible group structure multiplication: G x G --> G inversion: G --> G

21. Recap on Lie groups and Lie algebras Lie group G: analytic manifold with a compatible group structure multiplication: G x G --> G inversion: G --> G Example:

22. Recap on Lie groups and Lie algebras Lie group G: analytic manifold with a compatible group structure multiplication: G x G --> G inversion: G --> G Example: Lie algebra g : vector space with Lie bracket [x, y] = -[y,x] antisymmetry [a x + b y,z] = a [x, z] + b [y, z] linearity [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 Jacobi

23. Recap on Lie groups and Lie algebras Lie group G: analytic manifold with a compatible group structure multiplication: G x G --> G inversion: G --> G Example: Lie algebra g : vector space with Lie bracket [x, y] = -[y,x] antisymmetry [a x + b y,z] = a [x, z] + b [y, z] linearity [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 Jacobi Example:

24. Given a Lie group G its Lie algebra g is T e G. Adjoint and coadjoint action.

25. Given a Lie group G its Lie algebra g is T e G. Adjoint and coadjoint action. Example: G = SL(2, C ). Then

26. Given a Lie group G its Lie algebra g is T e G. Adjoint and coadjoint action. Example: G = SL(2, C ). Then g acts on itself by the adjoint action:

27. Given a Lie group G its Lie algebra g is T e G. Adjoint and coadjoint action. Example: G = SL(2, C ). Then g acts on itself by the adjoint action: g acts on g * by the coadjoint action:

28. Example: Symmetric non-degenerate bilinear form:

29. Example: Symmetric non-degenerate bilinear form: Coadjoint action:

30. Example: Symmetric non-degenerate bilinear form: Coadjoint action:

31. Example: Symmetric non-degenerate bilinear form: Coadjoint action:

32. Loop algebra

33. Commutator:

34. Loop algebra Commutator: Killing form:

35. Loop algebra Commutator: Killing form: Subalgebra:

36. Loop algebra Commutator: Killing form: Subalgebra: Dual space:

37. Loop algebra Commutator: Killing form: Subalgebra: Dual space:

38. Loop algebra Commutator: Killing form: Subalgebra: Dual space:

39. Coadjoint orbits

40. Integrable systems = flows on coadjoint orbits:

41. Coadjoint orbits Integrable systems = flows on coadjoint orbits: Example: PII

42. Coadjoint orbits Integrable systems = flows on coadjoint orbits: Example: PII

43. Coadjoint orbits Integrable systems = flows on coadjoint orbits: Example: PII

44. Kostant - Kirillov Poisson bracket on the dual of a Lie algebra

45. Differential of a function

46. Kostant - Kirillov Poisson bracket on the dual of a Lie algebra Differential of a function Example: PII. Take

47. Definition:

48. Example:

49. Definition: Example:

50. Definition: Example:

51. Definition: Example:

52. Hamiltonians

53. Fix a function

54. Hamiltonians Fix a function For every define:

55. Hamiltonians Fix a function For every define: Kostant Kirillov Poisson bracket:

56. Hamiltonians Fix a function For every define: Kostant Kirillov Poisson bracket: Define then we get the evolution equation: